## Living_dreams 3 years ago find all local max,min and/or saddle point of f(x,y)=4+x^3+y^-3xy

1. Avva

would you plz write the function using equation

2. amistre64

hmm, something to do with:\[f_{xy}f_{yx}-(f_{xx})^2\] or did i recall that incorrectly?

3. amistre64

4. Living_dreams

\[4+x ^{3}+y ^{3}-3xy\]

5. Living_dreams

it fxx.fyy-(fxy)^2

6. Living_dreams

i jst want to verify the answer for it !

7. TuringTest

8. Avva

\[fx = 2x-3y \] \[fy= 2y-3x\] on solving both equations the point is (0,0) \[fxx = 2\] , \[fyy= 2\] \[fxy = \] \[D = -5 <0 saddle point\]

9. TuringTest

that's not quite right I don't think

10. Avva

fxy = -3

11. TuringTest

\[f(x,y)=4+x^3+y^3-3xy\]\[f_x=3x^2-3y=0\implies y=x^2\]\[f_y=3y^2-3x=3x^4-3x=3x^3(x-1)=0\]\[ x=\{0,1\},y=\{0,1\}\]

12. Living_dreams

its x^3 so i have (0,0) ,(1,1)and a (-1,1) i am not sure if (-1,1) should be there oor not

13. Avva

@Living_dreams I solved what you mentioned in your comment

14. TuringTest

there is no x=-1 solution, I don't see how you get that

15. TuringTest

@Avva I don't see how you get your fx and fy

16. Avva

Ooh I saw the powers on X ^ y 2 not 3

17. Living_dreams

@Avva thanks for ur help but i was wonder abt the max and min points! n\ @TuringTest : 3y^2-3x=0 i sub y=x^2 from other equation so it will be 3x(x^3-1)=0 therefore you have x =0,1,-1 i am not sure abt -1 though !! tht was my question

18. TuringTest

x^3=1 does not have x=-1 as a solution since (-1)^3=-1, not 1

19. TuringTest

I did screw up my factoring earlier though, you were right about that :P

20. TuringTest

3x(x^3-1)=0 3x=0 -> x=0 x^3=1 -> x=1

21. Living_dreams

@TuringTest that alrite!! :) so now my question is x^3=1 so will i hv x=-1 if i solve for x ?

22. TuringTest

no, again, because \((-1)^3\neq1\) so \(x^3=1\) has only one real solution, x=1 x=-1 is NOT a solution

23. Avva

the other two roots are complex not real

24. Living_dreams

@TuringTest oh ok thanks a lot for ur help!! so the final ans will be (0,0 ) is a saddle point and (1,1) will be a local minimum !

25. TuringTest

exactly :)

26. Living_dreams

@Avva :thanks for ur help !! :)

27. Avva

@Living_dreams URW :) sorry I didn't get that the powers are 3 at first

28. Living_dreams

it okay !!i knw it get hard to tell the difference between 2 and 3 if we type it in equation form !!